# Solving Systems of Equations

• K. O. Geddes
• S. R. Czapor
• G. Labahn
Chapter

## Abstract

In this chapter we consider the classical problem of solving (exactly) a system of algebraic equations over a field F. This problem, along with the related problem of solving single univariate equations, was the fundamental concern of algebra until the beginning of the “modern” era (roughly, in the nineteenth century); it remains today an important, widespread concern in mathematics, science and engineering. Although considerable effort has been devoted to developing methods for numerical solution of equations, the develop- ment of exact methods is also well motivated. Obviously, exact methods avoid the issues of conditioning and stability. Moreover, in the case of nonlinear systems, numerical methods cannot guarantee that all solutions will be found (or prove that none exist). Finally, many systems which arise in practice contain “free” parameters and hence must be solved over non-numerical domains.

## Keywords

Integral Domain Computer Algebra Gaussian Elimination Common Root Univariate Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
E.H. Bareiss, “Sylvester’s Identity and Multistep Integer-Preserving Gaussian Elimination,” Math. Comp., 22(103) pp. 565–578 (1968).
2. 2.
E.H. Bareiss, “Computational Solutions of Matrix Problems Over an Integral Domain,” J. Inst. Maths Applcs, 10 pp. 68–104 (1972).
3. 3.
S. Cabay, “Exact Solution of Linear Equations,” pp. 392–398 in Proc. SYMSAM’ 71, ed. S.R. Petrick, ACM Press (1971).Google Scholar
4. 4.
S. Cabay and T.P.L. Lam, “Congruence Techniques for the Exact Solution of Integer Systems of Linear Equations,” ACM TOMS, 3(4) pp. 386–397 (1977).
5. 5.
J.F. Canny, E. Kaltofen, and L. Yagati, “Solving Systems of Non-Linear Polynomial Equations Faster,” pp. 121–128 in Proc. ISSAC’ 89, ed. G.H. Gonnet, ACM Press (1989).Google Scholar
6. 6.
G.E. Collins, “The Calculation of Multivariate Polynomial Resultants,” J. ACM, 18(4) pp. 515–532 (1971).
7. 7.
G.E. Collins, “Quantifier Elimination for Real Closed Fields: A Guide to the Literature,” pp. 79–81 in Computer Algebra — Symbolic and Algebraic Computation (Second Edition), ed. B. Buchberger, G.E. Collins and R. Loos, Springer-Verlag, Wien — New York (1983).Google Scholar
8. 8.
W.M. Gentleman and S.C. Johnson, “Analysis of Algorithms, A Case Study: Determinants of Matrices with Polynomial Entries,” ACM TOMS, 2(3) pp. 232–241 (1976).
9. 9.
M.L. Griss, “The Algebraic Solution of Sparse Linear Systems via Minor Expansion,” ACM TOMS, 2(1) pp. 31–49 (1976).
10. 10.
J.A. Howell and R.T. Gregory, “An Algorithm for Solving Linear Algebraic Equations using Residue Arithmetic I, II,” BIT, 9 pp. 200–224, 324–337 (1969).
11. 11.
D. Lazard, “Systems of Algebraic Equations,” pp. 88–94 in Proc. EUROSAM’ 79, Lecture Notes in Computer Science 72, ed. W. Ng, Springer-Verlag (1979).Google Scholar
12. 12.
J.D. Lipson, “Symbolic methods for the computer solution of linear equations with applications to flowgraphs,” pp. 233–303 in Proc. of the 1968 Summer Inst. on Symb. Math. Comp., ed. R. G. Tobey, (1969).Google Scholar
13. 13.
M.T. McClellan, “The Exact Solution of Systems of Linear Equations with Polynomial Coefficients,” J. ACM, 20(4) pp. 563–588 (1973).
14. 14.
T. Sasaki and H. Murao, “Efficient Gaussian Elimination Method for Symbolic Determinants and Linear Systems,” ACM TOMS, 8(3) pp. 277–289 (1982).
15. 15.
H. Takahasi and Y. Ishibashi, “A New Method for’ Exact Calculation’ by a Digital Computer,” Inf. Processing in Japan, 1 pp. 28–42 (1961).Google Scholar
16. 16.
B.L. van der Waerden, Modern Algebra (Vols. I and II), Ungar (1970).Google Scholar
17. 17.
D.Y.Y. Yun, “On Algorithms For Solving Systems of Polynomial Equations,” ACM SIGSAM Bull., 27 pp. 19–25 (1973).

• K. O. Geddes
• 1
• S. R. Czapor
• 2
• G. Labahn
• 1